Knives and other tools are commonly sharpened by inserting the blade between cooperating grinding wheels. Cup-shaped hollow cylinder grinding wheels are used to hollow grind blades and have certain advantages over solid cylindrical grinding wheels. Cup-shaped grinding wheels have an outer beveled grinding surface located at an angle of about 50.degree. to 60.degree. with respect to the axis of the wheel and the blade to be sharpened is inserted between the mating beveled surfaces of the grinding wheels. With the use of cup-shaped grinding wheels, the depth of grinding, that is, the distance from the cutting edge of the blade to the back edge, remains constant regardless of the wear on the wheels. This differs from use of solid cylindrical grinding wheels in which the depth of grinding will vary as the wheels are worn.
Periodically, grinding wheels must be dressed to remove metal deposits from the wheels and to correct out-of-roundness or eccentricity due to wear. Dressing of the beveled grinding surfaces of cup-shaped grinding wheels is difficult due to the fact that the beveled grinding surfaces are not in a common plane, except for the line of tangency where the surfaces mate, so that the dressing of the beveled surfaces has normally been done by individually dressing the wheels, as opposed to simultaneously dressing both wheels.